Search Results (81)

We experimentally and theoretically demonstrate coexistence of three different wavelength-multiplexed bound dual-frequency pulses in an all-fiber mode-locked fiber laser, effectively achieved by exploiting polarization-dependent loss effects and two uneven gain peaks of Er-doped fiber. With the single wall carbon-nanotube-based intensity modulation, wavelength-multiplexed dual-frequency pulses located at 1531.1 nm and 1556.6 nm are obtained. Changing the polarization rotation angles in the fiber cavity, one of the two asynchronous pulses evolves into a bound state of a doublet, in which the center wavelength of the bound solitons is centered at ~1530 nm or ~1556 nm. The relative phase between the two bound solitons or modulation depth of bound solitons can be switched by a polarization controller. A simulation method based on coupled Ginzburg–Landau equations is provided to characterize the laser physics and understand the mechanism behind the dynamics of tuning between different bound dual-frequency pulses. The proposed fiber laser will provide a potential way to understand multiple soliton dynamics and implementation in optical frequency combs generation. Full article

We present a linearly implicit and structure-preserving scheme to solve the space-fractional Ginzburg–Landau–Schrödinger equation. The fully discrete scheme is obtained by combining the modified leap-frog method in the temporal direction and the finite difference methods in the spatial direction. It is shown that the scheme can be unconditionally energy-stable. In particular, the equation becomes the space-fractional Schrödinger equation. Then, the scheme can keep both the discrete mass and energy conserved. Moreover, convergence of the scheme is obtained. Numerical experiments are performed to confirm the theoretical results. Full article

Lithium batteries have emerged as the mainstream technology in the current energy storage field due to their advantages, such as high energy density and long cycle life. However, from a multi-physics coupling perspective, research remains relatively scarce regarding the analysis of dendrite nucleation and growth, as well as their influence on lithium dendrite growth. Based on the phase field theory, this study develops a mechanical-thermal-electrochemical coupling model to systematically investigate the evolution mechanisms and suppression strategies of lithium dendrites induced by relevant physical quantities through the coupled effects of mechanical, thermal, and electrochemical fields. The dynamic behavior of the solid-solid interface is characterized by introducing order parameters. The governing nonlinear partial differential equations are formulated by combining the Cahn-Hilliard and Ginzburg-Landau equations. The present numerical results and the previous results are compared to validate the present model in properly predicting lithium dendrite growth. Numerical simulations are performed to analyze the influence of various physical parameters, such as electric potential, anisotropic intensity and anisotropic modulus, on the morphological evolution of lithium dendrites. These findings provide critical insights for advancing strategies to suppress lithium dendrite growth and enhance battery performance in solid-state lithium batteries under multi-field coupling conditions. Full article

The phase transitions and switching dynamics of topological polar textures in hafnium oxide (HfO₂)-based cylindrical-shell ferroelectrics are studied using a three-dimensional (3D) phase field model based on the self-consistent solution of the time-dependent Ginzburg–Landau model and Poisson equation. The comprehensive interplays of bulk free energy, gradient energy, depolarization energy, and elastic energy are taken into account. When a cylindrical ferroelectric device is biased under the in-plane radial electric field, there is a size-controlled phase transition between the ferroelectric (FE), antiferroelectric (AFE), and paraelectric (PE) phases, depending on ferroelectric film thickness and cylindrical shell radius. For in-plane polarization textures at the equilibriums, the FE phase has a Néel-like texture with a center-type four-quad domain, the AFE phase has a monodomain texture, and the PE phase has a Bloch-like texture with a vortex four-quad domain. These polarization domain textures are resultant from energy competition and topologically protected by the geometrical confinement. The polarization dynamics from polar states towards equilibriums are analyzed considering the separated contributions of x- and y-components of polarizations that are driven by x-y in-plane electric fields. The emergent topological domains and phase transitions provide guidelines for geometrical engineering of a novel nano-structured ferroelectric device that is different from the planar one, offering new possibilities for multi-functional high-density ferroelectric memory. Full article

This paper proposes a dual-fractional framework for stochastic differential equations (SDEs) that integrates conformable fractional calculus into both the system dynamics and the stochastic driving noise. For the first time, a conformable formulation of fractional noise is introduced, replacing the traditional Caputo-based representation. This modification eliminates singular kernel functions while preserving the fundamental properties of classical calculus, thereby simplifying both the analysis and numerical implementation. A complete analytical study is presented, rigorously addressing the convergence properties, deriving explicit error estimates, and establishing the numerical stability of the proposed scheme. The framework is realized through an enhanced conformable fractional discrete Temimi–Ansari method (CFDTAM), which accommodates distinct fractional orders for the system dynamics and the stochastic component. The stability and accuracy of the proposed scheme are validated through comparisons with the stochastic Runge–Kutta method (SRK) as implemented in Mathematica 12. Applications to benchmark models—including the fractional Langevin, Ginzburg–Landau, and Davis–Skodje systems—further demonstrate the robustness of the framework, especially in regimes where the Hurst exponent $\overline{\mathrm{Ӊ}}$ greater than 0.5. Overall, the results establish the method as a rigorous and efficient tool for modelling and analyzing stochastic fractional systems in finance, biophysics, and engineering. Full article

The fractal extension of the time-dependent Ginzburg–Landau (TDGL) equation, formulated within the framework of Scale Relativity, generalizes superconducting dynamics to non-differentiable space–time. Although analytically well established, its numerical solution remains difficult because of the strong coupling between amplitude and phase curvature. Here we develop two complementary deep learning solvers for the fractal TDGL (FTDGL) system. The Fractal Physics-Informed Neural Network (F-PINN) embeds the Scale-Relativity covariant derivative through automatic differentiation on continuous fields, whereas the Fractal Graph Neural Network (F-GNN) represents the same dynamics on a sparse spatial graph and learns local gauge-covariant interactions via message passing. Both models are trained against finite-difference reference data, and a parametric study over the dimensionless fractality parameter $\mathcal{D}$ quantifies its influence on the coherence length, penetration depth, and peak magnetic field. Across multivortex benchmarks, the F-GNN reduces the relative L₂ error on ${\left|\psi \right|}^2$ from 0.190 to 0.046 and on B_z from approximately 0.62 to 0.36 (averaged over three seeds). This ≈4× improvement in condensate-density accuracy corresponds to a substantial enhancement in vortex-core localization—from tens of pixels of uncertainty to sub-pixel precision—and yields a cleaner reconstruction of the 2π phase winding around each vortex, improving the extraction of experimentally relevant observables such as ξ_eff, λ_eff, and local B_z peaks. The model also preserves flux quantization and remains robust under 2–5% Gaussian noise, demonstrating stable learning under experimentally realistic perturbations. The $\mathcal{D}$—scan reveals broader vortex cores, a non-monotonic variation in the penetration depth, and moderate modulation of the peak magnetic field, while preserving topological structure. These results show that graph-based learning provides a superior inductive bias for modeling non-differentiable, gauge-coupled systems. The proposed F-PINN and F-GNN architectures therefore offer accurate, data-efficient solvers for fractal superconductivity and open pathways toward data-driven inference of fractal parameters from magneto-optical or Hall-probe imaging experiments. Full article

We experimentally and through simulations demonstrate a passively mode-locked fiber laser based on nonlinear polarization rotation, which generates the evolution from h-shaped pulses to GHz harmonic trains. When the polarization angle is continuously changed, the h-shaped pulse sequentially evolves into multiple pulses, bunched solitons, and harmonic pulses. The maximum order of harmonic trains obtained in experiments is 120, corresponding to the repetition frequency of 1.03996 GHz. The coupled Ginzburg-Landau equation and two-time-scale approach to gain is provided to characterize the laser physics. The fast and slow evolution of gain contributes to the stabilization and length of one soliton pattern, respectively. The proposed fiber laser is cost effective and easy to implement, providing a potential way to study soliton dynamics in depth. Full article

This study proposes a novel structure-preserving algorithm for the Ginzburg–Landau equation (GLE) by combining the Fourier pseudospectral method with the Exponential Average Vector Field (EAVF) scheme. The proposed numerical framework strictly preserves the energy dissipation property of GLE systems, as validated through theoretical analysis and numerical experiments on solitary wave dynamics. Compared to conventional methods such as the average vector field approach, the EAVF-based scheme demonstrates superior computational efficiency, including faster convergence and enhanced stability under larger time steps, enabling accurate long-term simulation of strongly nonlinear GLE systems. Furthermore, this research incorporates a pedagogical innovation through its implementation within an undergraduate innovation project. By adopting a “problem decomposition–code verification–modular development” training model, students engage in the full cycle of algorithm design, implementation, and validation. This practice-oriented approach significantly enhances students’ competencies in scientific programming, complex problem-solving, and research-oriented thinking, providing an effective paradigm for synergizing advanced computational research with talent cultivation in STEM education. Full article

This paper investigates the chaotic and pattern dynamics of the time-fractional Ginzburg–Landau equation. First, we propose a high-precision numerical method that combines finite difference schemes with an improved Grünwald–Letnikov fractional derivative approximation. Subsequently, the effectiveness of the proposed method is validated through systematic comparisons with classical numerical approaches. Finally, numerical simulations based on this method reveal rich dynamical phenomena in the fractional Ginzburg–Landau equation: the system exhibits complex behaviors including chaotic oscillations and novel two- and three-dimensional pattern structures. This study not only advances the theoretical development of numerical solutions for fractional GLE but also provides a reliable computational tool for deeper understanding of its complex dynamical mechanisms. Full article

In this article, we investigate the regularization and qualitative properties of parabolic Ginzburg–Landau equations in variable exponent Herz spaces. These spaces capture both local and global behavior, providing a natural framework for our analysis. We establish boundedness results for fractional Bessel–Riesz operators, construct examples highlighting their advantage over classical Riesz potentials, and recover several known theorems as special cases. As an application, we analyze a parabolic Ginzburg–Landau operator with VMO coefficients, showing that our estimates ensure the boundedness and continuity of solutions. Full article

It is well known that nonlinear partial differential equations (NLPDEs) can only be solved numerically and that fourth-order NLPDEs in their derivatives require unconventional methods. This paper explains spectral numerical methods for obtaining a numerical solution by Fast Fourier Transform (FFT), implemented under Python in tis version 3.1 and their libraries (NumPy, Tkinter). Examples of NLPDEs typical of Condensed Matter Physics to be solved numerically are the conserved Cahn–Hilliard, Swift–Hohenberg and conserved Swift–Hohenberg equations. The last two equations are solved by the first- and second-order exponential integrator method, while the first of these equations is solved by the conventional FFT method. The Cahn–Hilliard equation, a phase-field model with an extended Ginzburg–Landau-like functional, is solved in two-dimensional (2D) to reproduce the evolution of the microstructure of an amorphous alloy Ce₇₅Al_{25 − x}Gax, which is compared with the experimental micrography of the literature. Finally, three-dimensional (3D) simulations were performed using numerical solutions by FFT. The second-order exponential integrator method algorithm for the Swift–Hohenberg equation implementation is successfully obtained under Python by FFT to simulate different 3D patterns that cannot be obtained with the conventional FFT method. All these 2D/3D simulations have applications in Materials Science and Engineering. Full article

This paper provides highly dispersive optical soliton solutions to the perturbed complex Ginzburg–Landau equation. The self-phase modulation structures are maintained in three forms, which are derived from the power law of nonlinearity with arbitrary intensity. The paper employs the semi-inverse variational principle as its integration scheme, as conventional methods are incapable for it. The amplitude–width relation of the solitons is reconstructed by employing Cardano’s method to solve a cubic polynomial equation. Also presented are the necessary parameter constraints that naturally arise from the scheme. These findings enhance our understanding of soliton dynamics and pave the way for further research into more complex nonlinear systems. Future studies may explore the implications of these results in various physical contexts, potentially leading to novel applications in fields such as fiber optics and quantum fluid dynamics. Full article

This paper investigates quiescent solitons in optical fibers and crystals, modeled by the complicated Ginzburg–Landau equation incorporating nonlinear chromatic dispersion and six self-phase modulation structures introduced by Kudryashov. The model is formulated with linear temporal evolution and analyzed using Lie symmetry methods. The study also identified parameter constraints under which solutions exist. Full article

To improve the sampling efficiency of Markov Chain Monte Carlo in complex parameter spaces, this paper proposes an adaptive sampling method that integrates a swarm intelligence mechanism called the BeeSwarm-MH algorithm. The method combines global exploration by scout bees with local exploitation by worker bees. It employs multi-stage perturbation intensities and adaptive step-size tuning to enable efficient posterior sampling. Focusing on Bayesian inference for parameter estimation in the soliton solutions of the two-dimensional complex Ginzburg–Landau equation, we design a dedicated inference framework to systematically compare the performance of BeeSwarm-MH with the classical Metropolis–Hastings algorithm. Experimental results demonstrate that BeeSwarm-MH achieves comparable estimation accuracy while significantly reducing the required number of iterations and total computation time for convergence. Moreover, it exhibits superior global search capabilities and adaptive features, offering a practical approach for efficient Bayesian inference in complex physical models. Full article

The weakly nonlinear stability analysis of a convective flow in a planar vertical fluid layer is performed in this paper. The base flow in the vertical direction is generated by internal heat sources distributed within the fluid. The system of Navier–Stokes equations under the Boussinesq approximation and small-Prandtl-number approximation is transformed to one equation containing a stream function. Linear stability calculations with and without a small-Prandtl-number approximation lead to the range of the Prantdl numbers for which the approximation is valid. The method of multiple scales in the neighborhood of the critical point is used to construct amplitude evolution equation for the most unstable mode. It is shown that the amplitude equation is the complex Ginzburg–Landau equation. The coefficients of the equation are expressed in terms of integrals containing the linear stability characteristics and the solutions of three boundary value problems for ordinary differential equations. The results of numerical calculations are presented. The type of bifurcation (supercritical bifurcation) predicted by weakly nonlinear calculations is in agreement with experimental data. Full article